A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph

An autonomous AI research agent has reported a systematic attack on Conway's 99-graph problem, proving that no circulant graph on Z/99 satisfies more than 68.0% of the constraints (33 of 49 difference-classes), and achieving a best verified artifact at 69.43% with evidence of a robust frontier across fourteen methods. The agent also introduced a forced-structure reduction that collapses the existence problem to a 12-regular graph on 84 vertices, validated by recovering the unique srg(9,4,1,2).

arXiv:2608.11211v1 Announce Type: new Abstract: Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg} 99,14,1,2 $ exists. We report a systematic, fully reproducible attack by an autonomous AI research agent, scored under the track's partial-credit metric. Our verifiable contributions are: 1 an exhaustive proof that no circulant graph on $\mathbb{Z}/99$ satisfies more than $3366/4950=68.0\%$ of the constraints $33$ of $49$ difference-classes , with the same ceiling for the other abelian group of order $99$; 2 a forced-structure reduction: $\lambda=1$ makes each neighbourhood a perfect matching and $\mu=2$ puts the outer vertices in bijection with non-matched neighbour-pairs, collapsing existence to a $12$-regular graph on $84$ vertices, encoded for CP-SAT and validated by recovering the unique $\mathrm{srg} 9,4,1,2 $; 3 a validated prescribed-automorphism orbit-existence framework fixed-point-free and single-fixed-point actions, checked on $\mathrm{srg} 9,4,1,2 $ and the Paley graph $\mathrm{srg} 13,6,2,3 $ , and 4 a best verified artifact at $69.43\%$, with evidence that this is a robust frontier fourteen distinct methods, none exceeding it entangled with the open question, since any provable bound below $4950$ is a non-existence proof.